Do you suggest that Whitehead and Russell's statement of the principle of ''reductio ad absurdum'' is incorrect or invalid? See the statement of *2.01 on page 104 of Volume 1: [http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=AAT3201.0001.001;didno=AAT3201.0001.001;view=pdf;seq=00000126]. That is, (''x'' &rarr; ~''x'') &rarr; ~''x'' is a tautology and it is the application of this tautology which may be used. Thus, if one demonstrates ''x'' &rarr; ~''x'', one may use *2.01 together with the primitive proposition *1.1 on page 98 [http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=AAT3201.0001.001;didno=AAT3201.0001.001;view=pdf;seq=00000120] which states that 'Anything implied by a true elementary proposition is true' (''modus ponens'') to conclude that ~''x'' is true. Russell and Whitehead use principle to prove, for example, that it cannot be possible that (p = q) & (p = ~q) on page 129 [http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=AAT3201.0001.001;didno=AAT3201.0001.001;view=pdf;seq=00000151]. The equivalent statement (~p &rarr; p) &rarr; p is given in *2.18. In the given proof of there being infinitely many primes, there are ''N'' primes implies there are greater than ''N'' primes (or, in other words, there are not ''N'' primes); therefore, we conclude that there are not ''N'' primes for any integer value of ''N''. It does so by observing that the constructed number ''q'' must have a divisors other than the ones which were listed and that at least one of those divisors must have the properties of a prime number. This is in contrast to the [http://wiki.ironchariots.org/index.php?title=Reductio_ad_absurdum&oldid=11830 previous proof] which simply stated that that product plus one is prime which is false. The current version, however, is also a valid proof using ''modus tollens''. - [[User:Dwharder|Dwharder]] 9:00, 17 October 2010 (EDT)

Do you suggest that Whitehead and Russell's statement of the principle of ''reductio ad absurdum'' is incorrect or invalid? See the statement of *2.01 on page 104 of Volume 1: [http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=AAT3201.0001.001;didno=AAT3201.0001.001;view=pdf;seq=00000126]. That is, (''x'' &rarr; ~''x'') &rarr; ~''x'' is a tautology and it is the application of this tautology which may be used. Thus, if one demonstrates ''x'' &rarr; ~''x'', one may use *2.01 together with the primitive proposition *1.1 on page 98 [http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=AAT3201.0001.001;didno=AAT3201.0001.001;view=pdf;seq=00000120] which states that 'Anything implied by a true elementary proposition is true' (''modus ponens'') to conclude that ~''x'' is true. Russell and Whitehead use principle to prove, for example, that it cannot be possible that (p = q) & (p = ~q) on page 129 [http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=AAT3201.0001.001;didno=AAT3201.0001.001;view=pdf;seq=00000151]. The equivalent statement (~p &rarr; p) &rarr; p is given in *2.18. In the given proof of there being infinitely many primes, there are ''N'' primes implies there are greater than ''N'' primes (or, in other words, there are not ''N'' primes); therefore, we conclude that there are not ''N'' primes for any integer value of ''N''. It does so by observing that the constructed number ''q'' must have a divisors other than the ones which were listed and that at least one of those divisors must have the properties of a prime number. This is in contrast to the [http://wiki.ironchariots.org/index.php?title=Reductio_ad_absurdum&oldid=11830 previous proof] which simply stated that that product plus one is prime which is false. The current version, however, is also a valid proof using ''modus tollens''. - [[User:Dwharder|Dwharder]] 9:00, 17 October 2010 (EDT)

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: I would not suggest that R&W are wrong, just that it's not relevant to this article. I don't believe ''reductio ad absurdum'' arguments are ever actually presented in the form, "Assume P; deduce not P; conclude not P." Instead, they're of the form, "Assume P; deduce a contradiction [Q and not Q]; conclude not P." Your reworked proof, in fact, is of this form ("Assume there are finitely many prime numbers... We can therefore list all N prime numbers... we have found a prime number not appearing in our list... There must be infinitely many prime numbers"). That is exactly in the form "Assume P; deduce Q and not Q; conclude not P." - [[User:Dcljr|dcljr]] 04:26, 22 October 2010 (CDT)

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: I would not suggest that R&W are wrong, just that it's not relevant to this article. I don't believe ''reductio ad absurdum'' arguments are ever actually presented in the form, "Assume P; deduce not P; conclude not P." Instead, they're of the form, "Assume P; deduce a contradiction [i.e., Q and not Q]; conclude not P." Your [http://wiki.ironchariots.org/index.php?title=Reductio_ad_absurdum&diff=prev&oldid=14673 reworked proof] was, in fact, of this form ("Assume there are finitely many prime numbers... We can therefore list all N prime numbers... we have found a prime number not appearing in our list... There must be infinitely many prime numbers"). That is exactly in the form, "Assume P; deduce Q and not Q; conclude not P." - [[User:Dcljr|dcljr]] 04:26, 22 October 2010 (CDT)

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:: BTW, for the benefit of other readers, Dwharder is not responding to my objection in the [[#Not already listed?|previous section of this talk page]]; he's responding to my removal of the "Self contradiction" form of RAA [http://wiki.ironchariots.org/index.php?title=Reductio_ad_absurdum&diff=14679&oldid=14675 which he added to the article] after rewording the "primes" proof. Two separate issues. - [[User:Dcljr|dcljr]] 05:11, 22 October 2010 (CDT)

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: (To continue my reply...) Dwharder, you also said that my original proof "''simply stated that that product plus one is prime which is false''". Not quite. I ''concluded'' that it was prime ''because'' it was not divisible by any (other) prime number. I was relying on the same thing that makes [http://wiki.ironchariots.org/index.php?title=Reductio_ad_absurdum&oldid=14686 the current proof] work—that every composite number is divisible by some prime number, and hence if a number is not divisible by any (other) prime number, it must be prime—I just didn't state the justification explicitly. Now, your observation that not all serial-prime-product-plus-one numbers are prime (your example being "2×3×5×7×11×13 + 1 = 30031 = 51×509") is not relevant in the context of the original proof since clearly 51 is not one of our (presumed exhaustive) list of prime numbers (in your example the only primes would be 2, 3, 5, 7, 11, and 13)! In other words, your objection only "works" because one ''cannot'', in fact, create a finite list of all primes. You can't raise that objection in the midst of the proof, however, because by assumption there ''are no'' primes beyond the last multiplied number. See? - [[User:Dcljr|dcljr]] 05:11, 22 October 2010 (CDT)

Latest revision as of 05:22, 22 October 2010

Rolled back to previous version.

This is not a form of the straw man fallacy. Reductio ad absurdum is a logical argument which attempts to disprove a claim by assuming it as the major premise and demonstrating that the claim cannot be true by arriving at a false conclusion in a valid argument with a minor premise which is known to be true. The principle is that a logically valid syllogism is one where if both premises are true, the conclusion must be true. If the conclusion is false, one or more of the premises must be false. By demonstrating that the minor premise is true, the suspect premise must be false. - Sans Deity 20:48, 30 August 2006 (MST)

Umm... what "second premise"? The article only mentions a single premise, the one that ultimately gets rejected. - dcljr 01:58, 31 August 2006 (MST) Irrelevant now that above comment has been reworded. - dcljr 22:55, 1 September 2006 (MST)

Not already listed?

The part of the proof starting with "or it is divisible by a prime number which has not yet been listed" is not necessary since we already assumed we'd multiplied all the primes. It's like saying, "Assume 1, 2, and 3 are the only numbers. Now consider a new number, 4." If the premise is correct, then 4 must be one of the other three numbers and not something different! The previous version of the proof, while admittedly not as precise, didn't make any such immediately falsifiable statements along the way, and yet lead to an absurdity. I've reworded the proof yet again to be even more precise. The argument is now even closer to Euclid's (Elements, Book IX, Proposition 20). Unfortunately, it relies on a result that must be proved separately (Bk. VII, Prop. 31). Such is the cost of added precision... - dcljr 16:00, 16 October 2010 (CDT)

Whitehead and Russell's Principia Statement of Reductio ad Absurdum

Do you suggest that Whitehead and Russell's statement of the principle of reductio ad absurdum is incorrect or invalid? See the statement of *2.01 on page 104 of Volume 1: [1]. That is, (x → ~x) → ~x is a tautology and it is the application of this tautology which may be used. Thus, if one demonstrates x → ~x, one may use *2.01 together with the primitive proposition *1.1 on page 98 [2] which states that 'Anything implied by a true elementary proposition is true' (modus ponens) to conclude that ~x is true. Russell and Whitehead use principle to prove, for example, that it cannot be possible that (p = q) & (p = ~q) on page 129 [3]. The equivalent statement (~p → p) → p is given in *2.18. In the given proof of there being infinitely many primes, there are N primes implies there are greater than N primes (or, in other words, there are not N primes); therefore, we conclude that there are not N primes for any integer value of N. It does so by observing that the constructed number q must have a divisors other than the ones which were listed and that at least one of those divisors must have the properties of a prime number. This is in contrast to the previous proof which simply stated that that product plus one is prime which is false. The current version, however, is also a valid proof using modus tollens. - Dwharder 9:00, 17 October 2010 (EDT)

I would not suggest that R&W are wrong, just that it's not relevant to this article. I don't believe reductio ad absurdum arguments are ever actually presented in the form, "Assume P; deduce not P; conclude not P." Instead, they're of the form, "Assume P; deduce a contradiction [i.e., Q and not Q]; conclude not P." Your reworked proof was, in fact, of this form ("Assume there are finitely many prime numbers... We can therefore list all N prime numbers... we have found a prime number not appearing in our list... There must be infinitely many prime numbers"). That is exactly in the form, "Assume P; deduce Q and not Q; conclude not P." - dcljr 04:26, 22 October 2010 (CDT)

(To continue my reply...) Dwharder, you also said that my original proof "simply stated that that product plus one is prime which is false". Not quite. I concluded that it was prime because it was not divisible by any (other) prime number. I was relying on the same thing that makes the current proof work—that every composite number is divisible by some prime number, and hence if a number is not divisible by any (other) prime number, it must be prime—I just didn't state the justification explicitly. Now, your observation that not all serial-prime-product-plus-one numbers are prime (your example being "2×3×5×7×11×13 + 1 = 30031 = 51×509") is not relevant in the context of the original proof since clearly 51 is not one of our (presumed exhaustive) list of prime numbers (in your example the only primes would be 2, 3, 5, 7, 11, and 13)! In other words, your objection only "works" because one cannot, in fact, create a finite list of all primes. You can't raise that objection in the midst of the proof, however, because by assumption there are no primes beyond the last multiplied number. See? - dcljr 05:11, 22 October 2010 (CDT)